1 Uninformed Search (Ch. 3-3.4)
7 Small examples 8-Queens: how to fit 8 queens on a 8x8 board so no 2 queens can capture each other Two ways to model this: Incremental = each action is to add a queen to the board (1.8 x 10 14 states) Complete state formulation = all 8 queens start on board, action = move a queen (2057 states)
8 Real world examples Directions/traveling (land or air) Model choices: only have interstates? Add smaller roads, with increased cost? (pointless if they are never taken)
10 Real world examples Traveling salesperson problem (TSP): Visit each location exactly once and return to start Goal: Minimize distance traveled
11 Search algorithm To search, we will build a tree with the root as the initial state Any problems with this?
12 Search algorithm
13 Search algorithm 8-queens can actually be generalized to the question: Can you fit n queens on a z by z board? Except for a couple of small size boards, you can fit z queens on a z by z board This can be done fairly easily with recursion (See: nqueens.py)
14 Search algorithm We can remove visiting states multiple times by doing this: But this is still not necessarily all that great...
15 Search algorithm Next we will introduce and compare some tree search algorithms These all assume nodes have 4 properties: 1. The current state 2. Their parent state (and action for transition) 3. Children from this node (result of actions) 4. Cost to reach this node (from root)
16 Search algorithm When we find a goal state, we can back track via the parent to get the sequence To keep track of the unexplored nodes, we will use a queue (of various types) The explored set is probably best as a hash table for quick lookup (have to ensure similar states reached via alternative paths are the same in the hash, can be done by sorting)
17 Search algorithm The search algorithms metrics/criteria: 1. Completeness (does it terminate with a valid solution) 2. Optimality (is the answer the best solution) 3. Time (in big-o notation) 4. Space (big-o) b = maximum branching factor d = minimum depth of a goal m =maximum length of any path(depth of tree)
18 Uninformed search Today, we will focus on uninformed search, which only have the node information (4 parts) (the costs are given and cannot be computed) Next time we will continue with informed searches that assume they have access to additional structures of the problem (i.e. if costs were distances between cities, you could also compute the distance as the bird flies )
19 Breadth first search Breadth first search checks all states which are reached with the fewest actions first (i.e. will check all states that can be reached by a single action from the start, next all states that can be reached by two actions, then three...)
(see: https://www.youtube.com/watch?v=5ufmu9tsoem) (see: https://www.youtube.com/watch?v=ni0dt288vls) 20 Breadth first search
21 Breadth first search BFS can be implemented by using a simple FIFO (first in, first out) queue to track the fringe/frontier/unexplored nodes Metrics for BFS: Complete (i.e. guaranteed to find solution if exists) Non-optimal (unless uniform path cost) Time complexity = O(b d ) Space complexity = O(b d )
22 Breadth first search Exponential problems are not very fun, as seen in this picture:
23 Uniform-cost search Uniform-cost search also does a queue, but uses a priority queue based on the cost (the lowest cost node is chosen to be explored)
24 Uniform-cost search The only modification is when exploring a node we cannot disregard it if it has already been explored by another node We might have found a shorter path and thus need to update the cost on that node We also do not terminate when we find a goal, but instead when the goal has the lowest cost in the queue.
25 Uniform-cost search UCS is.. 1. Complete (if costs strictly greater than 0) 2. Optimal However... 3&4. Time complexity = space complexity = O(b 1+C*/min(path cost) ), where C* cost of optimal solution (much worse than BFS)
26 Depth first search DFS is same as BFS except with a FILO (or LIFO) instead of a FIFO queue
27 Depth first search Metrics: 1. Might not terminate (not correct) (e.g. in vacuum world, if first expand is action L) 2. Non-optimal (just... no) 3. Time complexity = O(b m ) 4. Space complexity = O(b*m) Only way this is better than BFS is the space complexity...
29 Depth limited search DFS by itself is not great, but it has two (very) useful modifications Depth limited search runs normal DFS, but if it is at a specified depth limit, you cannot have children (i.e. take another action) Typically with a little more knowledge, you can create a reasonable limit and makes the algorithm correct
30 Depth limited search However, if you pick the depth limit before d, you will not find a solution (not correct, but will terminate)
31 Iterative deepening DFS Probably the most useful uninformed search is iterative deepening DFS This search performs depth limited search with maximum depth 1, then maximum depth 2, then 3... until it finds a solution
32 Iterative deepening DFS
33 Iterative deepening DFS The first few states do get re-checked multiple times in IDS, however it is not too many When you find the solution at depth d, depth 1 is expanded d times (at most b of them) The second depth are expanded d-1 times (at most b 2 of them) Thus
34 Iterative deepening DFS Metrics: 1. Complete 2. Non-optimal (unless uniform cost) 3. O(b d ) 4. O(b*d) Thus IDS is better in every way than BFS (asymptotically) Best uninformed we will talk about
35 Bidirectional search Bidirectional search starts from both the goal and start (using BFS) until the trees meet This is better as 2*(b d/2 ) < b d (the space is much worse than IDS, so only applicable to small problems)
36 Uninformed search